\babel@toc {ngerman}{}\relax 
\contentsline {section}{\numberline {.}Anmerkungen (DG)}{4}{section.0}%
\contentsline {subsubsection}{\numberline {.0.a.}\textbf {Hinweis an den Leser}}{5}{subsubsection.0.0.1}%
\contentsline {subsection}{\numberline {.1.}\textbf {Abschnitt 0. Vorbemerkungen}}{6}{subsection.0.1}%
\contentsline {subsubsection}{\numberline {.1.a.}\textbf {Empfohlene Literatur}}{6}{subsubsection.0.1.1}%
\contentsline {subsubsection}{\numberline {.1.b.}\textbf {Notation}}{6}{subsubsection.0.1.2}%
\contentsline {section}{\numberline {I.}\textbf {I. Kapitel: Grundlagen}}{10}{section.1}%
\contentsline {subsection}{\numberline {I.1.}\textbf {Abschnitt 1. Moduln, Algebren, Tensorprodukte, Kategorien, Funktoren}}{10}{subsection.1.1}%
\contentsline {subsubsection}{\numberline {I.1.a.}\textbf {Fundamentale Eigenschaften von Moduln}}{10}{subsubsection.1.1.1}%
\contentsline {subsubsection}{\numberline {I.1.b.}\textbf {Tensorielle Abbildungen und Tensorprodukte}}{14}{subsubsection.1.1.2}%
\contentsline {subsubsection}{\numberline {I.1.c.}\textbf {Zusatzstruktur auf Tensorprodukten}}{19}{subsubsection.1.1.3}%
\contentsline {subsubsection}{\numberline {I.1.d.}\textbf {Tensorprodukte \"{u}ber einem K\"{o}rper}}{28}{subsubsection.1.1.4}%
\contentsline {subsubsection}{\numberline {I.1.e.}\textbf {Tensorprodukte von }$n$ \textbf {Moduln}}{29}{subsubsection.1.1.5}%
\contentsline {subsubsection}{\numberline {I.1.f.}\textbf {Zusatzstruktur auf }$\operatorname *{Hom}$}{32}{subsubsection.1.1.6}%
\contentsline {subsubsection}{\numberline {I.1.g.}\textbf {Algebren}}{34}{subsubsection.1.1.7}%
\contentsline {subsubsection}{\numberline {I.1.h.}\textbf {Kategorien und Funktoren}}{43}{subsubsection.1.1.8}%
\contentsline {subsubsection}{\numberline {I.1.i.}\textbf {Nat\"{u}rliche Transformationen zwischen Funktoren}}{49}{subsubsection.1.1.9}%
\contentsline {subsubsection}{\numberline {I.1.j.}\textbf {Adjungierte Funktoren}}{52}{subsubsection.1.1.10}%
\contentsline {subsubsection}{\numberline {I.1.k.}\textbf {Quasiinverse \"{A}quivalenzen}}{57}{subsubsection.1.1.11}%
\contentsline {subsection}{\numberline {I.2.}\textbf {Abschnitt 2. Coalgebren, Bialgebren, Hopfalgebren}}{60}{subsection.1.2}%
\contentsline {subsubsection}{\numberline {I.2.a.}\textbf {Coalgebren}}{61}{subsubsection.1.2.1}%
\contentsline {subsubsection}{\numberline {I.2.b.}\textbf {Summenlose} \textbf {Sweedler-Notation}}{87}{subsubsection.1.2.2}%
\contentsline {subsubsection}{\numberline {I.2.c.}\textbf {Eindeutigkeit der Coeins}}{91}{subsubsection.1.2.3}%
\contentsline {subsubsection}{\numberline {I.2.d.}\textbf {Faltung und Algebrastruktur auf }$\operatorname *{Hom}\left ( C,A\right ) $}{95}{subsubsection.1.2.4}%
\contentsline {subsubsection}{\numberline {I.2.e.}\textbf {``Fast''-Dualit\"{a}t zwischen Algebren und Coalgebren}}{98}{subsubsection.1.2.5}%
\contentsline {subsubsection}{\numberline {I.2.f.}\textbf {Gruppen\"{a}hnliche und primitive Elemente}}{102}{subsubsection.1.2.6}%
\contentsline {subsubsection}{\numberline {I.2.g.}\textbf {Bialgebren}}{107}{subsubsection.1.2.7}%
\contentsline {subsubsection}{\numberline {I.2.h.}\textbf {Hopfalgebren}}{109}{subsubsection.1.2.8}%
\contentsline {subsubsection}{\numberline {I.2.i.}\textbf {Eigenschaften der Antipode}}{111}{subsubsection.1.2.9}%
\contentsline {subsubsection}{\numberline {I.2.j.}\textbf {Quotienten von Bialgebren und Hopfalgebren}}{122}{subsubsection.1.2.10}%
\contentsline {subsubsection}{\numberline {I.2.k.}\textbf {Beispiele f\"{u}r Hopfalgebren}}{127}{subsubsection.1.2.11}%
\contentsline {subsubsection}{\numberline {I.2.l.}$\operatorname *{op}$ \textbf {und }$\operatorname *{cop}$}{140}{subsubsection.1.2.12}%
\contentsline {subsubsection}{\numberline {I.2.m.}\textbf {Eine andere Charakterisierung von Hopfalgebren}}{144}{subsubsection.1.2.13}%
\contentsline {subsubsection}{\numberline {I.2.n.}\textbf {Moduln \"{u}ber Hopfalgebren}}{150}{subsubsection.1.2.14}%
\contentsline {subsubsection}{\numberline {I.2.o.}\textbf {Graduierte Strukturen}}{160}{subsubsection.1.2.15}%
\contentsline {subsubsection}{\numberline {I.2.p.}\textbf {Zusammenh\"{a}ngende graduierte Bialgebren}}{163}{subsubsection.1.2.16}%
\contentsline {subsubsection}{\numberline {I.2.q.}\textbf {Die Tensorhopfalgebra}}{173}{subsubsection.1.2.17}%
\contentsline {subsubsection}{\numberline {I.2.r.}\textbf {Die Shufflehopfalgebra}}{189}{subsubsection.1.2.18}%
\contentsline {subsection}{\numberline {I.3.}\textbf {Abschnitt 3. }$H$\textbf {-Modulalgebren und Smash-Produkt}}{192}{subsection.1.3}%
\contentsline {subsubsection}{\numberline {I.3.a.}$H$\textbf {-Modulalgebren}}{192}{subsubsection.1.3.1}%
\contentsline {subsubsection}{\numberline {I.3.b.}\textbf {Smash-Produkte}}{195}{subsubsection.1.3.2}%
\contentsline {subsection}{\numberline {I.4.}\textbf {Abschnitt 4. Comoduln und }$H$\textbf {-Comodulalgebren}}{202}{subsection.1.4}%
\contentsline {subsubsection}{\numberline {I.4.a.}\textbf {Comoduln}}{202}{subsubsection.1.4.1}%
\contentsline {subsubsection}{\numberline {I.4.b.}\textbf {Der Endlichkeitssatz}}{210}{subsubsection.1.4.2}%
\contentsline {subsubsection}{\numberline {I.4.c.}$H$\textbf {-Comodulalgebren}}{212}{subsubsection.1.4.3}%
\contentsline {subsubsection}{\numberline {I.4.d.}\textbf {Adjungierte Modul- und Comodulstrukturen}}{215}{subsubsection.1.4.4}%
\contentsline {subsubsection}{\numberline {I.4.e.}\textbf {Direkte Summen von Comoduln}}{226}{subsubsection.1.4.5}%
\contentsline {subsubsection}{\numberline {I.4.f.}\textbf {Direkte Summen von Coalgebren}}{231}{subsubsection.1.4.6}%
\contentsline {subsection}{\numberline {I.5.}\textbf {Abschnitt 5. Affine Schemata und affine Gruppen}}{232}{subsection.1.5}%
\contentsline {subsubsection}{\numberline {I.5.a.}\textbf {Das Yoneda-Lemma}}{232}{subsubsection.1.5.1}%
\contentsline {subsubsection}{\numberline {I.5.b.}\textbf {Affine Schemata und affine Gruppen}}{235}{subsubsection.1.5.2}%
\contentsline {subsection}{\numberline {I.6.}\textbf {Abschnitt 6. Einschub: Einige Resultate aus der Algebra}}{250}{subsection.1.6}%
\contentsline {subsubsection}{\numberline {I.6.a.}\textbf {Das Jacobson-Radikal}}{250}{subsubsection.1.6.1}%
\contentsline {subsubsection}{\numberline {I.6.b.}\textbf {Der Satz von Krull-Remak-Schmidt}}{264}{subsubsection.1.6.2}%
\contentsline {subsubsection}{\numberline {I.6.c.}\textbf {K\"{u}rzung f\"{u}r }$A$\textbf {-Linksmoduln}}{265}{subsubsection.1.6.3}%
\contentsline {subsubsection}{\numberline {I.6.d.}\textbf {K\"{u}rzung f\"{u}r mehrfache Linksmoduln}}{273}{subsubsection.1.6.4}%
\contentsline {section}{\numberline {II.}\textbf {II. Kapitel: Struktur cokommutativer Hopfalgebren}}{278}{section.2}%
\contentsline {subsection}{\numberline {II.1.}\textbf {Abschnitt 1. Liealgebren und ihre universellen Einh\"{u}llenden}}{278}{subsection.2.1}%
\contentsline {subsubsection}{\numberline {II.1.a.}\textbf {Liealgebren}}{278}{subsubsection.2.1.1}%
\contentsline {subsubsection}{\numberline {II.1.b.}\textbf {Die universelle Einh\"{u}llende einer Liealgebra}}{298}{subsubsection.2.1.2}%
\contentsline {subsection}{\numberline {II.2.}\textbf {Abschnitt 2. Satz von Poincar\'{e}-Birkhoff-Witt}}{304}{subsection.2.2}%
\contentsline {subsubsection}{\numberline {II.2.a.}\textbf {Einige Eigenschaften in Bezug auf Untercoalgebren}}{319}{subsubsection.2.2.1}%
\contentsline {subsection}{\numberline {II.3.}\textbf {Abschnitt 3. Irreduzible cokommutative Hopfalgebren in Charakteristik }$0$}{327}{subsection.2.3}%
\contentsline {subsection}{\numberline {II.4.}\textbf {Abschnitt 4. Struktur cokommutativer Hopfalgebren in Charakteristik }$0$}{336}{subsection.2.4}%
\contentsline {subsection}{\numberline {II.5.}\textbf {Abschnitt 5. Hilfss\"{a}tze \"{u}ber die duale Algebra einer Coalgebra}}{346}{subsection.2.5}%
\contentsline {subsubsection}{\numberline {II.5.a.}\textbf {Lemmata aus der Linearen Algebra}}{346}{subsubsection.2.5.1}%
\contentsline {subsubsection}{\numberline {II.5.b.}\textbf {Anwendung auf die duale Algebra}}{360}{subsubsection.2.5.2}%
\contentsline {subsubsection}{\numberline {II.5.c.}\textbf {Die Coradikalfiltrierung: Lemmata}}{368}{subsubsection.2.5.3}%
\contentsline {subsubsection}{\numberline {II.5.d.}\textbf {Die Coradikalfiltrierung: Beweis von 4.7}}{371}{subsubsection.2.5.4}%
\contentsline {subsubsection}{\numberline {II.5.e.}\textbf {Die Coradikalfiltrierung: Weitere Eigenschaften}}{375}{subsubsection.2.5.5}%
\contentsline {subsection}{\numberline {II.6.}\textbf {Ausblick aufs zweite Semester}}{376}{subsection.2.6}%
\contentsline {subsubsection}{\numberline {II.6.a.}\textbf {Ausblick}}{376}{subsubsection.2.6.1}%
\contentsline {section}{\numberline {III.}\textbf {III. Kapitel: Endlichdimensionale Hopfalgebren}}{377}{section.3}%
\contentsline {subsection}{\numberline {III.1.}\textbf {Abschnitt 1. Hopfmoduln und Integrale}}{377}{subsection.3.1}%
\contentsline {subsubsection}{\numberline {III.1.a.}\textbf {Hopfmoduln und ihre coinvarianten Elemente}}{377}{subsubsection.3.1.1}%
\contentsline {subsubsection}{\numberline {III.1.b.}\textbf {Integrale}}{381}{subsubsection.3.1.2}%
\contentsline {subsubsection}{\numberline {III.1.c.}\textbf {Frobeniusalgebren}}{388}{subsubsection.3.1.3}%
\contentsline {subsubsection}{\numberline {III.1.d.}\textbf {Endlichdimensionale Hopfalgebren als Frobeniusalgebren}}{395}{subsubsection.3.1.4}%
\contentsline {subsection}{\numberline {III.2.}\textbf {Abschnitt 2. Die Ordnung der Antipode}}{395}{subsection.3.2}%
\contentsline {subsection}{\numberline {III.3.}\textbf {Abschnitt 3. Halbeinfache Hopfalgebren}}{401}{subsection.3.3}%
\contentsline {subsection}{\numberline {III.4.}\textbf {Abschnitt 4. Die S\"{a}tze von Nichols-Zoeller und Skryabin}}{426}{subsection.3.4}%
\contentsline {subsubsection}{\numberline {III.4.a.}\textbf {Einf\"{u}hrendes Beispiel: Gruppenalgebren}}{426}{subsubsection.3.4.1}%
\contentsline {subsubsection}{\numberline {III.4.b.}\textbf {Linkscoidealunteralgebren vs. Coideale-und-Rechtsideale}}{430}{subsubsection.3.4.2}%
\contentsline {subsubsection}{\numberline {III.4.c.}\textbf {Hilfsmittel aus der Algebra}}{433}{subsubsection.3.4.3}%
\contentsline {subsubsection}{\numberline {III.4.d.}\textbf {Costabile Ideale und Rechtscomodulunteralgebren}}{436}{subsubsection.3.4.4}%
\contentsline {subsubsection}{\numberline {III.4.e.}\textbf {Anwendung auf die Quotiententheorie}}{445}{subsubsection.3.4.5}%
\contentsline {subsection}{\numberline {III.5.}\textbf {Abschnitt 5. Darstellungen halbeinfacher Hopfalgebren}}{455}{subsection.3.5}%
\contentsline {subsubsection}{\numberline {III.5.a.}\textbf {Einfache }$H$\textbf {-Linksmoduln und das Schur-Lemma}}{455}{subsubsection.3.5.1}%
\contentsline {subsubsection}{\numberline {III.5.b.}\textbf {Der Hopfalgebrenfall}}{462}{subsubsection.3.5.2}%
\contentsline {section}{\numberline {IV.}\textbf {IV. Kapitel: Drinfeld-Doppel und Bosonisierung}}{462}{section.4}%
\contentsline {subsection}{\numberline {IV.1.}\textbf {Abschnitt 1. }$2$\textbf {-Cozykeltwist}}{462}{subsection.4.1}%
\contentsline {subsection}{\numberline {IV.2.}\textbf {Abschnitt 2. Das Drinfeld-Doppel}}{477}{subsection.4.2}%
\contentsline {subsection}{\numberline {IV.3.}\textbf {Abschnitt 3. Einschub: Duale Coalgebren}}{485}{subsection.4.3}%
\contentsline {subsubsection}{\numberline {IV.3.a.}\textbf {Die duale Coalgebra }$A^{0}$}{485}{subsubsection.4.3.1}%
\contentsline {subsubsection}{\numberline {IV.3.b.}\textbf {Eine Verallgemeinerung}}{489}{subsubsection.4.3.2}%
\contentsline {subsection}{\numberline {IV.4.}\textbf {Abschnitt 4. Drinfeld-Doppel und quasitriangul\"{a}re Hopfalgebren}}{494}{subsection.4.4}%
\contentsline {subsubsection}{\numberline {IV.4.a.}\textbf {Quasitriangul\"{a}re Hopfalgebren}}{494}{subsubsection.4.4.1}%
\contentsline {subsection}{\numberline {IV.5.}\textbf {Abschnitt 5. Yetter-Drinfeld-Moduln}}{501}{subsection.4.5}%
\contentsline {subsection}{\numberline {IV.6.}\textbf {Abschnitt 6. Bosonisierung}}{502}{subsection.4.6}%